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Notes on Ultrasound - Echocardiography
51:060 Fundamentals of Bioimaging
Edwin L. Dove
1412 Seamans Center
335-5635
Edwin-dove@uiowa.edu
Table of Contents
1. Physics of Ultrasound............................................................................... 1
1.1 Wave propagation ..................................................................................................... 1
1.2 Generation of pressure wave ..................................................................................... 8
1.3 Wave reflection ....................................................................................................... 10
2. Medical Ultrasound ................................................................................ 13
3. Clinical Ultrasound ................................................................................ 15
Reference:...................................................................................................... 18
1. Physics of Ultrasound
1.1 Wave propagation
By definition, ultrasound is sound having a frequency greater than 20,000 cycles per
second, that is, sound above the audible range. Actually, frequencies in the range of
millions of cycles per second are used for medical diagnostic purposes. There are at least
four advantages of ultrasound energy in medical diagnostic imaging: 1) ultrasound can be
directed in a beam, 2) ultrasound obeys the laws of reflection and refraction, 3)
ultrasound energy is reflected off of small objects, and 4) ultrasound has no known
deleterious health effects. There are two principal disadvantages of ultrasound. The first
is that the energy propagates poorly through a gaseous medium. It is virtually impossible
for ultrasound to pass through air; thus ultrasound transducers must have airless contact
with the body during examinations. In addition, it is difficult to examine parts of the
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body that contain air. The second disadvantage is that ultrasound images are relatively
noisy, and have poorer contrast than X-ray and MR modalities.
Acoustic (sound) waves are merely organized vibrations of the molecules or atoms of a
medium that is able to support the propagation of these waves. Usually the vibrations are
organized in a sinusoidal fashion. Error! Reference source not found. demonstrates that
a sinusoidal sound wave is actually a series of areas of compression and rarefactions.
Such changes are frequently depicted or described as a sine wave with the peak of the
“hill” representing the pressure maximum, and the nadir of the “valley” representing the
pressure minimum. The combination of one compression and one rarefaction represents
one cycle. The distance between the peak of one compression to the peak of the next is a
wavelength.
Figure 1.1 – Illustration of areas of compression and rarefaction due to sinusoidal application of
pressure.
The areas of compression and rarefaction are due to periodic pressure being applied to the
surface of the medium, which in our case is human tissue.
In order to understand how images are formed, and to design proper equipment and
algorithms, we need to develop an appropriate mathematical model. To do this, consider
the incremental volume element shown in Figure 1.2 representing a small section of
human tissue.
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Figure 1.2 Pressures and velocities in an incremental volume of tissue.
Newton’s force equation yields
F = ma = m
du
 ∂u ∂u ∂z 
= m +

dt
 ∂t ∂z ∂t 
(1.1)
which becomes
∂u 
 ∂u
F = m + u 
∂z 
 ∂t
(1.2)
Since pressure is force per unit area, the net force on the volume is
F = [ p( z ) − p( z + ∆z )] A
(1.3)
then Equation (1.2) becomes (after substituting for mass as m = ρ A∆z )
p ( z ) = p ( z + ∆z )
∂u 
 ∂u
= ρ +u 
∆z
∂z 
 ∂t
(1.4)
Taking the limit as ∆z → 0 yields
−
∂p
∂u 
 ∂u
= ρ +u 
∂z
∂z 
 ∂t
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(1.5)
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Let’s assume that the density has an average (unperturbed) value ρ 0 and a time-varying
part ρ1 . Let’s further assume that ρ = ρ0 + ρ1 , where ρ1 ρ0 . Furthermore, the
quantities in Equation (1.4) represent small variations, so that whenever two of these
variables appear together as a product in an equation, then this product is second order
and can be neglected when compared to other first-order terms in which there is only one
of these variables. Using this reasoning, Equation (1.5) can be reduced to
∂p
∂u
+ ρ0
=0
∂z
∂t
(1.6)
Equation (1.6) has two unknowns, which means that this equation cannot be solved. To
find a an additional equation (and therefore a solution), let’s rely on a conservation of
mass analysis. This says that the net mass that leaves the incremental volume must equal
the net mass that enters the volume. The net mass that leaves the volume per unit time is
given by A [ ρ ( z + ∆z )u ( z + ∆z ) − ρ ( z )u ( z ) ] and this must be matched by a rate of change
of mass given by − A∆z [ ∂ρ ∂t ] . Equating these two quantities leads to
ρ ( z + ∆z )u ( z + ∆z ) − ρ ( z )u ( z )
∂ρ
=−
∆z
∂t
(1.7)
Taking the limit as ∆z → 0 yields the differential equation
∂ρ u ∂ρ
+
=0
∂z∂y ∂t
(1.8)
Using the same reasoning as before, this yields
ρ0
∂u ∂ρ1
+
=0
∂z ∂t
(1.9)
To simplify, let’s define the adiabatic compressibility constant K as
ρ1
= Kp
ρ0
(1.10)
and substituting this into Equation (1.10) yields
∂p 1 ∂u
+
=0
∂t K ∂z
(1.11)
Equations (1.11) and (1.6) represent a coupled pair of equations that must be solved
simultaneously. In this case it is actually easier to combine these into one equation as:
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∂2 p
∂ 2u
+ ρ0
=0
∂z 2
∂z∂t
(1.12)
and by taking the partial derivative of Equation (1.11) and combining, we get the
universally famous and ubiquitous wave equation
∂2 p
∂2 p
−
K
=0
ρ
0
∂z 2
∂t 2
(1.13)
A similar equation can be derived for particle velocity.
One possible solution for the wave equation is
p = p+ cos(ω t − kz )
(1.14)
which is valid if the following holds:
k 2 = ρ0 Kω 2
(1.15)
This is the well-known dispersion equation, which I am sure that all engineers talk
about in glorious detail at parties and social events to which engineers are invited at most
one time.
Acoustic waves are attenuated in tissue due to divergence of the wavefront, elastic
reflection at planar surfaces, elastic scattering from irregular surfaces, or absorption of
the wave energy. The major cause of attenuation is the last, absorption of the wave
energy by the tissue. The absorbved energy is converted into heat, but since the energy
delivered in a diagnostic ultrasound procedure is so small, the heat generated does not
cause any known tissue damage.
From Morse and Ingard (Morse, P. and K. Ingard, Theroetical Acoustics, New York:
Mcgraw-Hill, 1968.), if the media is lossy, then the wave equation becomes
3
∂ 2 p  4η
∂2 p
 ∂ p
′
+
+
−
=0
η
k
ρ
k
0
 ∂z 2∂t
∂z 2  3
∂t 2

(1.16)
where as before k (m-1) is the propagation constant, ρ0 (kg·m3) is the average density of
the medium, and η and η ′ are measures of tissue viscosity (modeled as viscous fluids).
Equation (1.16) has a solution given by
p = Pm exp ( −α z ) cos (ω t − kz )
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(1.17)
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where Pm is the maximum pressure, and α is the attenuation coefficient. We will come
back to this equation later.
The wavelength λ (m) of the sound wave shown in Figure 1.1 is given by
λ=
2π
k
(1.18)
The velocity c (m·s-1) representing the speed at which sound waves travel through a
particular medium, is given by
c=
ω 2π f
=
k
k
(1.19)
where ω and f represent the frequency (in rad·s-1 and Hz) of the energy delivered to the
tissue. Combining Equations (1.18) and (1.19), the speed of sound in the medium is
c=λf
(1.20)
Thus for a particular medium, the frequency and wavelength are inversely related; the
higher the frequency, the shorter the wavelength.
The speed of sound in a medium depends on the average density ρ0 (kg·m3) and the
compressibility K (m2·N-1) of the material (or tissue in our case). The relationship is
given by
c=
1
ρ0 K
(1.21)
Engineers and scientists working in ultrasound have found that a convenient way of
expressing relevant tissue properties is to use characteristic (or acoustic) impedance Z
(kg·m-2 ·s-1), which is defined by
Z = ρ0 c
(1.22)
The relationship in Equation (1.22) is important to remember because it describes how
sound travels through a medium.
A graphical illustration of how the loss factor introduced into the wave equation affects
its magnitude is given in Figure 1.3. The exp(−α z ) term represents the exponential
decay in the envelope of the pressure wave’s amplitude as a function of distance. The
radiated power will decrease even faster. The power I is given by
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I=
p+ −2α z
e
cos 2 (ω t − kz ) = I + e −2α z cos 2 (ω t − kz )
Z
(1.23)
Thus the power exponentially decays at a rate of 2α with distance. The average power
will also decay at a rate of 2α .
Figure 1.3 Exponential decrease in the amplitude of the pressure wave (top panel) and of the power
density (bottom panel). The decreases are described by the linear attenuation coefficient.
Biological tissues are not ideal viscous fluids, so the relationship in Equation (1.23) does
not strictly hold. Viscosity is one of a class of phenomenon known as relaxation effects,
which means that the effect has an associated time constant and corresponding relaxation
frequency. This means that the viscosity term is not a constant but will be a function of
frequency. Theory shows (see Morse and Ingard) that the attenuation coefficient α is
expressed as
( )
 4η + η ′ ω 2


3
α=
3
2 ρ0c
(1.24)
Table 1.1 contains typical values of acoustic parameters for selected human tissues. (this
is Table 4.3 in Christensen)
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1.2 Generation of pressure wave
The use of ultrasound became practical with the development of piezoelectric crystal
transducers. “Piezo” means pressure, so piezoelectric means that pressure is generated
when electrical energy is applied to the crystal. The quartz crystal in your watch is an
example of this material. When electrical energy is applied to the face of the crystal, the
shape of the crystal changes with the polarity of the electrical energy. As the crystal
expands and contracts it produces compressions and rarefactions, or sound waves. Figure
1.4 shows a sketch of a crystal that has piezoelectric properties.
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Figure 1.4 – Illustration of effect of changing the polarity of electrical excitation applied to a
piezoelectric crystal. Note the deformation due to the change in excitation.
The crystal changes shape as the surrounding electrical field is reversed. The wavelength
of the emitted ultrasound is a function of the size of the crystal. Thus, the size of the
crystal determines the required frequency of excitation.
Piezoelectric crystals also have the property that if they are compressed, they produce an
electrical voltage. Thus, a piezoelectric crystal can produce a pulse of mechanical energy
(pressure pulse) by electrically exciting the crystal (i.e., they can be transmitters), and
they can produce a pulse of electrical energy by mechanically exciting the crystal (i.e.,
they can be receivers).
Commercial transducers use ceramics, such as barium titanate or lead zirconate titanate,
as the piezoelectric element. Figure 1 shows how the piezoelectric element is packaged
in a commercial transducer.
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Figure 1.5 – Schematic drawing of a typical ultrasound probe used in medical imaging.
The piezoelectric element is backed with some material that absorbs sound energy
directed backwards. This improves the shape of the forward energy.
1.3 Wave reflection
As sound travels through a medium, it essentially travels in a straight line. When the
beam reaches an interface between two media with different acoustic impedances, it
undergoes reflection and refraction. Figure 1 demonstrates the principle of reflection and
refraction.
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Figure 1.6 – Illustration of reflection and refraction of an acoustic wave encountering a boundary
between two types of tissue with different acoustic impedances.
As the sound travels through a relatively homogeneous medium (side #1), it propagates in
essentially a straight line. When the sound reaches an interface (side #2), part of the
incident beam is reflected, and part is refracted (transmitted). This is the principle of
almost all medical ultrasound technology.
The amount of sound that is reflected depends on the degree of difference between the
two media; the greater the acoustic mismatch, the greater the amount of sound reflected.
In addition, the amount of ultrasound reflected or refracted depends on the angle at which
the ultrasound beam hits the interface between the different media. As the angle of
incidence approaches 90°, a higher percentage of the ultrasound is reflected.
Mathematically, Snell’s law (from electromagnetic theory - Physics II) states
sinθ i λ1 c1
=
=
sinθ t λ2 c2
(1.25)
where the subscript i represents incident, and t represents transmitted (or refracted) (see
Figure 1). This assumes that the angle of incidence θ i equals the angle of reflectance θ r .
The reflectivity, defined as the ratio of the reflected pressure to the incident pressure, can
be derived as
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Z2
Z
_ 1
p
cosθ t cosθ i
R= r =
Z2
Z
pi
+ 1
cosθ t cosθ i
(1.26)
Note that at normal incidence, θ i = θ r =0. Thus the reflectivity in Equation (1.26)
becomes
R=
Z 2 − Z1
Z2 + Z 1
(1.27)
Figure 1 shows two types of echoes that can result in ultrasound imaging.
Figure 1.7 – Illustration of specular and scattered echoes.
The first, specular echoes, originate from relatively large, strongly reflective, regularly
shaped objects with smooth surfaces. These reflections are angle dependent, and are
described by Equation (1.26). This type of reflection is called specular reflection.
The second type of echoes are scattered that originate from small, weakly reflective,
irregularly shaped objects, and are less angle-dependent and less intense. Unfortunately,
the mathematical treatment of non-specular reflection (sometimes called “speckle”) is a
bit more complicated, and involves the Rayleigh probability density function, which we
will not have time to cover. This type of reflection, however, sometimes dominates
medical images, as you will see in the laboratory demonstrations.
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2. Medical Ultrasound
Table 2.1 gives the reflectivity R (Equation (1.27) at normal incidence for a variety of
tissue interfaces.
Table 2.1 – Reflectivity of Normally Incident Waves for Various Interfaces.
Materials at Interface
Brain-skull bone
Fat-muscle
Fat-kidney
Muscle-blood
Soft tissue-water
Soft tissue-air
Reflectivity
0.66
0.10
0.08
0.03
0.05
0.9995
Here we see that the interfaces between soft tissues have a reflectivity of under 0.10,
representing less than 1% of the energy being reflected. Other interfaces, such as
between tissue and bone, tissue and air, and tissue and the transducer, have strong
reflections.
Table 2.2 lists the average velocity c (Equation (1.21)) for various materials and tissue
types.
Table 2.2 – Propagation Velocity for Various Tissue Types
Tissue
Air
Fat
Human tissue (mean)
Brain
Blood
Skull bone
Water
Mean Velocity (m·s-1)
330
1450
1540
1541
1570
4080
1480
Fortunately, although various materials exhibit profound changes in their acoustic
velocity, the soft tissue of the body are limited to a range of about +5%.
The basis for ultrasound imaging in the reflection mode of the human body is the
assumption of a constant propagation velocity throughout the body. Most makers of
ultrasound systems assume a mean velocity of 1540 m·s-1. Error! Reference source not
found. shows a drawing of an ideal sound beam traveling in tissue with a 1500 m·s-1
velocity.
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Figure 2.1 – Drawing of ideal sound beam traveling in tissue with a velocity of 1500 m·sec-1.
The timing between the initiation of the pressure pulse (due to an electrical pulse) and the
returned reflected pulse yields the depth of the interface from the transducer surface,
which in this case is 20 cm.
As stated before, in human tissue, the energy is lost through scattering and absorption
phenomena. Absorption of acoustic energy increases the temperature of the tissue. This
phenomenon, known as thermal radiation, has been used with some limited success to
treat cancerous lesions in the breast and prostate gland. Most engineers and scientists
working in the ultrasound area don’t actually quantify attenuation with the η term, but
rather attenuation in human tissue is characterized as the “half-value layer,” or the “halfpower distance.” These terms refer to the distance that ultrasound will travel in a
particular tissue before its amplitude or energy is attenuated to half its original value.
Table 2. gives the half-power for tissues and substances important in echocardiography.
Table 2.1 – Half-power Distances for Tissues and Substances Important in Echocardiography
Material
Water
Blood
Soft tissue
except muscle
Bone
Air
Lung
Half–power distance (cm)
380
15
5 to 1
1 to 0.6
0.7 to 0.2
0.08
0.05
These values depend on the frequency of excitation; the values in Table 2.3 are measured
for a transducer excited at 2 MHz. Note that ultrasound energy can travel in water 380
cm before its power decreases to half of its original value. Attenuation is greater in soft
tissue, and even greater in muscle. Thus, a thick muscled chest wall will offer a
significant obstacle to the transmission of ultrasound. Non-muscle tissue such as fat
does not attenuate acoustic energy as much. The half-power distance for bone is still less
than muscle, which explains why bone is such a barrier to ultrasound. Air and lung tissue
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have extremely short half-power distances and represent severe obstacles to the
transmission of acoustic energy.
Clinically, one always remembers that the attenuation coefficient is essentially doubled
when the frequency is doubled.
3. Clinical Ultrasound
Clinical ultrasound machines present the image data in one of three (or four if the
machine is modern) modes. The first is the A-mode, in which the amplitude of the
reflections is plotted against time (or distance assuming that the velocity of sound in the
body is 1540 m·s-1). A-mode ultrasound is used to judge the depth of an organ, or
otherwise assess an organ’s dimensions. A-mode is has been used extensively in midline
echoencephalography and ophthalmologic scanning. Elements of an A-mode instrument
are illustrated in Figure 3.1.
Figure 3.1 Elements of an A-mode pulse echo instrument.
The details of the arrival times of echoes from the interior tissue boundaries are
illustrated in Figure 3.2. The boundary depths can be related to echo arrival times
through the proportionality constant 2/c.
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Figure 3.2 Illustration of arrival times in A-mode ultrasonography.
The electronics of the receiver system tailors the received signal for maximum
readability. This ‘tailoring” includes a time gain control (TGC) that helps compensate
for the ever-decreasing signal strengths from deeper tissues due to the greater attenuation
over longer paths. The rate of TGC increase should be approximately 1 dB/MHz for each
interval of time that corresponds to 1 cm of travel. For example, assuming c=1.54x105
cm s-1, the gain should increase at a rate of approximately 154 dB ms-1.
B-mode ultrasound produces the sector (or pie segment-shaped) scans that one normally
sees. B-model machines represent the vast majority of machines in existence. These
machines are used in echocardiology, obstetrical scans, abdominal scans, gynecological
scans, etc. These scans require either a mechanical scanner transducer (the transducer
moves to produce the sector scan), or a transducer with a linear array operated as a
phased array (as discussed in class). Figure 3.3 illustrates the components of a B-mode
instrument.
Figure 3.3 A mechanical scanned B-mode ultrasound system. The transducer is rocked back and
forth to produce a corresponding sector scan as illustrated.
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Instead of using a mechanical scanned probe, modern machines use an electronically
scanned phased array of piezoelectric probes. This is illustrated in Figure 3.4 in which
the direction of the ultrasound beam is altered by phasing the excitation of the array
elements.
Figure 3.4 Phase array antenna examples.
M-mode is used for analyzing moving body parts (the M-mode is sometimes called the
“Time Motion” or TM mode). M-mode is particularly useful for imaging moving heart
valves. This mode is illustrated in figure 3.5.
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Figure 3.5 An M-mode instrument. The transducer beam is stationary while the echoes from a
moving reflector are received at varying times.
Other derived modes are possible. One particularly useful mode is the C-mode in which
an image is derived by either placing a receiver opposite a transmitter (and ignoring
reflections), or by deriving the cross-section C-model data from three-dimensional
imaging (see PowerPoint presentation for an example).
Most modern machines can generate multiple modes, and the choice is left to the user.
Most modern machines can also display Doppler echo pictures in which the data being
imaged are due to Doppler shifts resulting from moving parts. This is particularly useful
for imaging blood flow.
Reference:
1. Christensen, D. Ultrasound Bioinstrumentation. Wiley, 1988.
2. Feigenbaum, H. Echocardiography. (3rd edition) Lea-Febiger, 1981.
3. Carr, J. and J. Brown. Introduction to Biomedical Equipment Technology. PrenticeHall, 1998.
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